Category Archives: abstractmath.org

Posts about the abstractmath.org website.

abstractmath.org, language, language of math, math, representations, understanding math

Mathematical definitions

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The definition of a concept in math has properties that are different from definitions in other subjects:

• Every correct statement about the concept follows logically from its definition.
• An example of the concept fits all the requirements of the definition (not just most of them).
• Every math object that fits all the requirements of the definition is an example of the concept.
• Mathematical definitions are crisp, not fuzzy.
• The definition gives a small amount of structural information and properties that are enough to determine the concept.
• Usually, much else is known about the concept besides what is in the definition.
• The info in the definition may not be the most important things to know about the concept.
• The same concept can have very different-looking definitions. It may be difficult to prove they give the same concept.
• Math texts use special wording to give definitions. Newcomers may not understand that the intent of an assertion is that it is a definition.

How many college math teachers ever explain these things?

I will expand on some of these concepts in future posts.

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abstractmath.org, exposition, Handbook, language of math, math, proof, understanding math

Abstractmath.org after four years

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I have been working on the abstractmath website for about four years now (with time off for three major operations). Much has been written, but there are still lots of stubs that need to be filled in. Also much of it needs editing for stylistic uniformity, and for filling in details and providing more examples in some hastily written sections that read like outlines. Not to mention correcting errors, which seem to multiply when I am not looking. The website consists of four main parts and some ancillary chapters. I will go into more detail about some of the parts in later articles.

The languages of math. This is a description of mathematical English and the symbolic language of math (which are two different languages!) with an emphasis on the problems they cause people new to abstract math (roughly, math after calculus). At this point, I have completed a fairly thorough edit of the whole chapter that makes it almost presentable. Start with the Introduction.

Proofs. Mathematical proofs are a central problem for abstract math newbies. People interested in abstract math must learn to read and understand proofs. A proof is narrated in mathematical English. A proof has a logical structure. The reader must extract the logical structure from the narrative form. The chapter on proofs gives examples of proofs and discusses the logical structure and its relationship with the narration. The introduction to the chapter on proofs tells more about it.

Understanding math. There are certain barriers to understanding math that are difficult to get over. Mathematicians, math educators and philosophers work on various aspects of these problems and this chapter draws on their work and my own observations as a mathematician and a teacher.

All true statements about a math object must follow from the definition. That sounds clear enough. But in fact there are subtleties about definitions teachers may not tell students about because they are not aware of them themselves. For example, a definition can really mislead you about how to think about a math object.

The section on math objects breaks new ground (in my opinion) about how to think about them. I also discuss representations and models and images and metaphors (which I think is especially important), and in shorter articles about other topics such as abstraction and pattern recognition.

Doing math. This chapter points out useful behaviors and dysfunctional behaviors in doing math, with concrete examples. Beginners need to be told that when proving an elementary theorem they need to rewrite what is to be proved according to the definitions. Were you ever told that? (If you went to a Jesuit high school, you probably were.) Beginners need to be told that they should not try the same computational trick over and over even though it doesn’t work. That they need to look at examples. That they need to zoom in and out, looking at a detail and then the big picture. We need someone to make movies illustrating these things.

These other articles are outside the main organization:
Topic articles. Sets, real numbers, functions, and so on. In each case I talk just a bit about the topic to get the newbie over the initial hump.
Diagnostic examples. Examples chosen to evoke a misunderstanding, with a link to where it is explained. This needs to be greatly expanded.
Attitudes. This explains my point of view in doing abstractmath.org. I expect to rewrite it.

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abstractmath.org, language, math

Parameters

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I have been putting off writing the section on parameters in abstractmath.org because I don’t very well understand the connections between the different ways the word is used.

A parametrized family of functions, say f_a(x) = x^2+a, is easily handled by using the adjunction that makes the category of sets cartesian closed. That is, the three ways of looking at it, function of x parametrized by a, a function of a parametrized by x, and a function of two variables, are all naturally equivalent in a completely satisfactory way.

When you talk about taking a curve defined by an equation, e.g. x^2 + y^2 =1, and parametrizing it, that is a different situation; I have never thought about it from a categorical point of view but it is not hard to explain at the level of abstractmath.org.

What I don’t understand is if there is any real connection between these two meanings. I would welcome comments!

There are other meanings of “parameter”. One is the usage, “An important parameter of a finite group is its order.” I am not sure how often this occurs, and I look at the usage with some disapproval. The order of a finite group is more like an invariant (e.g. under isomorphism) than a parameter. Now do I have to write about invariants, too? Ye gods, do I have to do everything myself?

Another usage is in programming languages, where it can be analyzed simply as a variable. But they have different kinds of parameters that are implemented differently.

What I really need to do is spend a day searching for the word in JStor. I swore when I started doing abstractmath.org I wasn’t going to do any more lexicographical research (after the Handbook.). But maybe looking around for a few hours won’t hurt.

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abstractmath.org, astounding math, math, proof

Writing Astounding Math Stories

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I have written a second Astounding Math Story, this one about factoring integers and primality testing, published here. (I announced ASM here.)

These stories are aimed at people interested in math but not very far along in studying abstract math, the same audience that my abstractmath website is concerned with. That makes the stories hard to write.

For one thing, they need to be streamlined as much as possible. Each story should explain one astounding fact clearly, making as few mathematical demands on the reader as possible. I have put in links, mostly to Wikipedia, to explain concepts used in the story. This story has lots of footnotes telling about further developments and fine points.

Another point is that it is sometimes hard to convince students that they need to be astounded! I mentioned the phenomenon that primality testing is faster than factorization many times in my teaching (mostly to computing science students). Often I had to work hard to get them to realize that there was something shocking about this: Being a composite means having proper factors, but the fast ways of discovering compositeness tell you it is composite without giving any clue as to what the proper factors are. Research mathematicians are familiar with the idea of proving something exists without being able to say what it is, but students often have to be led by the nose to grasp this idea.

With this story I have experimented with making it a dialog. That makes it easier to write about a conflict between new ideas and old presuppositions. I would love to get comments about how these stories are written as well as the math involved.

Experienced research mathematicians will probably not be Astounded by these stories. But the ones I am writing about have often given mathematicians in previous centuries a lot of trouble — they seemed unbelievable or contradictory. And modern students have trouble with these ideas, too. In the current ASM I mention Kronecker’s problem with nonconstructive existence proofs.

One of the worst problems comes with infinite decimal expansions of real numbers (which I intend to write about). You can prove that 1.000… – .999… but the students don’t really believe it. That may be because they don’t really believe that all the decimal digits are really there. (A lot of philosophers don’t believe this either, but almost all research mathematicians talk and act as if they do.)

By the way, it is amazing how often there is an article in Wikipedia that says just what you want the reader to know (and of course usually a lot more) and does it pretty well. Lately I have run across just one exception, the articles on context. I have been writing about context in mathematical writing for abmath (don’t look, it isn’t there yet) and have wound up going into more detail than I wanted because I could not refer to Wikipedia.

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